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Full text of "Treatise On Analysis Vol-Ii"

390       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

continuity to jg^(K, ju) ((13.9.12) and (13.9.13)). Hence, by (13.17.1), h is
bounded in measure with respect to JLI, and we may assume that h e
For all s, t e ^C(K) we have

(VT(s) - a | T(t) - a) = (T(s)V  a | T(0 - a) =    s?

-/

since the vectors T^y) * # form a total set in E, it follows that V = T(/z).

(iii) To say that F is stable with respect to T means that the ortho-
gonal projector PF commutes with T(u) for all we ^C(K) (15.5.3); hence
PF e r(*c(K)) by virtue of (ii), and is of the form T((px) by virtue of (i).

We can also characterize all the totalizing vectors of the given representa-
tion:

(15.10.7) For a vector g e L(K, /j) to Z> totalizing for the representation
MP 0/^c(K), it is necessary and sufficient that g(t) ^ 0 almost everywhere with
respect to /j.

In order that the classes (fg)~ , where/ runs through #C(K)> should gen-
erate a subspace which is not dense in the Hilbert space L(K, p), it is neces-
sary and sufficient that there should exist a nonnegligible h e -S?c(K, ft) such
that jfyhdn = Q for all/e<ifc(K) (6.3.1). Also we have $* e J&f (K, /i)
(13.11.7); hence the measure (gh)  jj, is zero, which implies that g(t)h(t) = 0
almost everywhere with respect to n (13.14.4). But since by hypothesis the
integrable set A of points t e K at which h(t) + 0 is not negligible, we must
have g(t) = 0 almost everywhere in A. This proves the proposition.

To such a totalizing vector g for M^ corresponds by definition (15.10.1)
the positive measure \g\2 - n on K. Since \g\2 is /z-integrable and ^0 almost
everywhere, we obtain in this way all the positive measures on K which are
equivalent to \JL (13.15.6). In other words, the measure \JL of the statement of
(15.10.1) is determined only up to equivalence by the representation T.

(15.10.8) Now let us consider an arbitrary nondegenerate representation Tof
^C(K) in a separable Hilbert space. E. From (15.5.6) we know that E is the
Hilbert sum of a sequence (En) of closed subspaces stable with respect to T,
and such that the restriction Tn of T to EB admits a totalizing vector an , for
each n. The definition of the measures fj,Xty given in (15.10.3) applies without a) = (T(wuv)  0| a) =